## 1. Introduction

## 2. Open Markov Processes

**open Markov process**, or open, continuous time, discrete state Markov chain, is a triple $(V,B,H)$ where V is a finite set of

**states**, $B\subseteq V$ is the subset of

**boundary states**and $H:{\mathbb{R}}^{V}\to {\mathbb{R}}^{V}$ is an

**infinitesimal stochastic Hamiltonian**

**population**at the $i\text{th}$ state. We call the resulting function $p:V\to [0,\infty )$ the

**population distribution**. Populations evolve in time according to the

**open master equation**

**steady state**distribution is a population distribution which is constant in time:

**closed Markov process**, or continuous time, discrete state Markov chain, is an open Markov process whose boundary is empty. For a closed Markov process, the open master equation becomes the usual master equation

**equilibrium**. We say an equilibrium $q\in {[0,\infty )}^{V}$ of a Markov process is

**detailed balanced**if

**open detailed balanced Markov process**is an open Markov process $(V,B,H)$ together with a detailed balanced equilibrium $q:V\to (0,\infty )$ on V. In Section 5, we define the “dissipation”, which depends on the detailed balanced equilibrium populations, hence we equip an open Markov process with a specific detailed balanced equilibrium of the underlying closed Markov process. Thus, if a Markov process admits multiple detailed balanced equilibria, we choose a specific one. Note that we consider only detailed balanced equilibria such that the populations of all states are non-zero. Later, it will become clear why this is important.

**net flow**of population from the $j\text{th}$ state to the $i\text{th}$ is

**net inflow**${J}_{i}\left(p\right)\in \mathbb{R}$ of a particular state to be

**Kolmogorov’s criterion**[2], namely that

**non-equilibrium steady state**is a steady state in which the net flow between at least one pair of states is non-zero. Thus, there could be population flowing between pairs of states, but in such a way that these flows still yield constant populations at all states. In a closed Markov process, the existence of non-equilibrium steady states requires that the rates of the Markov process violate Kolmogorov’s criterion. We show that open Markov processes with constant boundary populations admit non-equilibrium steady states even when the rates of the process satisfy Kolmogorov’s criterion. Throughout this paper, we use the term equilibrium to mean detailed balanced equilibrium.

## 3. Membrane Diffusion as an Open Markov Process

**boundary populations**. Given the values of ${p}_{A}$ and ${p}_{C}$, the steady state population ${p}_{B}$ compatible with these values is

## 4. The Category of Open Detailed Balanced Markov Processes

**finite set with populations**, i.e., a finite set X together with a map ${p}_{X}:X\to [0,\infty )$ assigning a population ${p}_{i}\in [0,\infty )$ to each element $i\in X$. A morphism $M:(X,{p}_{X})\to (Y,{p}_{Y})$ consists of an open detailed balanced Markov process $(V,B,H,q)$ together with

**input**and

**output**maps $i:X\to V$ and $o:Y\to V$ which

**preserve population**, i.e., ${p}_{X}=q\circ i$ and ${p}_{Y}=q\circ i$. The union of the images of the input and output maps form the boundary of the open Markov processes $B=i\left(X\right)\cup o\left(Y\right)$.

## 5. Principle of Minimum Dissipation

**Definition 1.**

**dissipation functional**of a population distribution p to be

**Definition 2.**

**principle of minimum dissipation with boundary population**$\mathit{b}$ if p minimizes $D\left(p\right)$ subject to the constraint that ${p|}_{b}=b$.

**Theorem 3.**

**Proof.**

**Definition 4.**

**steady state population-flow pair**if the flows arise from a population distribution which obeys the principle of minimum dissipation.

**Definition 5.**

**behavior**of an open detailed balanced Markov process with boundary B is the set of all steady state population-flow pairs $({p}_{B},{J}_{B})$ along the boundary.

## 6. Dissipation and Entropy Production

**relative entropy**of two distributions $p,q$ is given by

**thermodynamic flux**from j to i and

**thermodynamic force**. This quantity:

**molar fraction**of the $i\text{th}$ species with ${n}_{i}$ giving the number of moles of the $i\text{th}$ species [13]. Note that this is equal to the fraction of the population in the $i\text{th}$ state ${x}_{i}=\frac{{n}_{i}}{{\sum}_{i}{n}_{i}}=\frac{{p}_{i}}{{\sum}_{i}{p}_{i}}$. The difference in chemical potential between two states gives the force associated with the flow which seeks to reduce this difference in chemical potential

## 7. Minimum Dissipation versus Minimum Entropy Production

## 8. Discussion

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Another layer of membrane whose interior population is labeled by D and whose exterior populations are labeled by ${C}^{\prime}$ and E.

**Figure 6.**Composition of open detailed balanced Markov processes results in an open detailed balanced Markov process.

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